Past exam of the mathematical methods II course of the University of Oxford / 2019 / 9 / b / Solution

A dispersive medium has frequency-dependent phase velocity, so different Fourier components generally propagate with different phases. A dispersion relation specifies $\omega$ as a function of the wave number $k$, or an equivalent relation between them. For a monochromatic factor $e^{i(kx-\omega t)}$, a point of fixed phase moves atFor a narrow wave packet centered at $k_{\ast }$, expand $\omega (k)\simeq \omega (k_{\ast })+(k-k_{\ast }){\omega }^{\prime }(k_{\ast })$. The envelope then translates at the group velocityThe phase velocity tracks crests of a monochromatic wave. The group velocity tracks a slowly varying envelope when the narrow-band approximation is valid; in the ideal lossless guide here it also equals the longitudinal energy-transport speed. It is not a universal formula for signal-front speed in every possible dispersive medium.

For the propagating branch of the supplied electromagnetic waveguide relation, take ${\omega }_0>0$, $k>0$ and $\omega =\sqrt{{\omega }_0^2+c^2{k}^2}$. Differentiation givesThe product alone would also allow $v_p=v_g=c$; the strictly positive cutoff is what makes the inequalities strict. At $k=0$ the phase-speed expression is undefined, and these formulas are interpreted through the propagating limit.

A moving monochromatic crest does not carry a new message. An information-bearing modulation is made from a range of frequencies; here its packet and energy move below $c$, and the high-frequency limit approaches $c$. Causal propagation of a newly switched-on disturbance is governed by the electromagnetic equations, so the superluminal phase velocity does not permit faster-than-light signalling. These are the waveguide phase and group velocities for a mode with nonzero cutoff, rather than a zero-cutoff mode with both velocities equal to $c$.